Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
Plant food. A farmer can buy two types of plant food, mix A and mix B. Each cubic yard of mix A contains 20 pounds of phosphoric acid, 30 pounds of nitrogen, and 5 pounds of potash. Each cubic yard of mix B contains 10 pounds of phosphoric acid, 30 pounds of nitrogen, and 10 pounds of potash. The minimum monthly requirements are 460 pounds of phosphoric acid, 960 pounds of nitrogen, and 220 pounds of potash. If x is the number of cubic yards of mix A used and y is the number of cubic yards of mix B used, write a system of linear inequalities that indicates appropriate restraints on x and y. Find the set of feasible solutions graphically for the amounts of mix A and mix B that can be used.Nutrition. A dietician in a hospital is to arrange a special diet using two foods. Each ounce of food M contains 30 units of calcium, 10 units of iron, and 10 units of vitamin A. Each ounce of food N contains 10 units of calcium, 10 units of iron, and 30 units of vitamin A. The minimum requirements in the diet are 360 units of calcium, 160 units of iron, and 240 units of vitamin A. If x is the number of ounces of food M used and y is the number of ounces of food N used, write a system of linear inequalities that reflects the conditions indicated. Find the set of feasible solutions graphically for the amount of each kind of food that can be used.Psychology. A psychologist uses two types of boxes when studying mice and rats. Each mouse spends 10 minutes per day in box A and 20 minutes per day in box B. Each rat spends 20 minutes per day in box A and 10 minutes per day in box B. The total minimum time available per day is 800 minutes for box A and 640 minutes for box B. If x is the number of mice used and y the number of rats used, write a system of linear inequalities that indicates appropriate restrictions on x and y. Find the set of feasible solutions graphically.A manufacturing plant makes two types of inflatable boats––a two-person boat and a four-person boat. Each two-person boat requires 0.9 labor-hour from the cutting department and 0.8 labor-hour from the assembly department. Each four-person boat requires 1.8 labor-hours from the cutting department and 1.2 labor-hours from the assembly department. The maximum labor-hours available per month in the cutting department and the assembly department are 864 and 672, respectively. The company makes a profit of $25 on each two-person boat and $40 on each four-person boat. (A) Identify the decision variables. (B) Summarize the relevant material in a table similar to Table 1 in Example 1. (C) Write the objective function P. (D) Write the problem constraints and nonnegative constraints. (E) Graph the feasible region. Include graphs of the objective function for P=$5,000,P=$10,000,P=$15,000, and P=$21,600. (F) From the graph and constant-profit lines, determine how many boats should be manufactured each month to maximize the profit. What is the maximum profit?Refer to the feasible region S shown in Figure 3. (A) Let P=x+y. Graph the constant-profit lines through the points 5,5 and 10,10. Place a straightedge along the line with the smaller profit and slide it in the direction of increasing profit, without changing its slope. What is the maximum value of P ? Where does this maximum value occur? (B) Repeat part (A) for P=x+10y. (C) Repeat part (A) for P=10x+y.In Example 2B we saw that there was no optimal solution for the problem of maximizing the objective function z over the feasible region S. We want to add an additional constraint to modify the feasible region so that an optimal solution for the maximization problem does exist. Which of the following constraints will accomplish this objective? Ax20By4CxyDyx(A) Maximize and minimize z=4x+2y subject to the constraints given in Example 2A. (B) Maximize and minimize z=20x+5y subject to the constraints given in Example 2B.A chicken farmer can buy a special food mix A at 20c per pound and a special food mix B at 40c per pound. Each pound of mix A contains 3,000 units of nutrient N1 and 1,000 units of nutrient N2 ; each pound of mix B contains 4,000 units of nutrient N1 and 4,000 units of nutrient N2. If the minimum daily requirements for the chickens collectively are 36,000 units of nutrient N1 and 20,000 units of nutrient N2, how many pounds of each food mix should be used each day to minimize daily food costs while meeting (or exceeding) the minimum daily nutrient requirements? What is the minimum daily cost? Construct a mathematical model and solve using the geometric method.In Problem 1-8, if necessary, review Theorem 1. In Problems 1-4, the feasible region is the set of points on and inside the rectangle with vertices 0,0,12,0,0,5, and 12,5. Find the maximum and minimum values of the objective function Q over the feasible region. Q=7x+14yIn Problem 1-8, if necessary, review Theorem 1. In Problems 1-4, the feasible region is the set of points on and inside the rectangle with vertices 0,0,12,0,0,5, and 12,5. Find the maximum and minimum values of the objective function Q over the feasible region. Q=3x+15yIn Problem 1-8, if necessary, review Theorem 1. In Problems 1-4, the feasible region is the set of points on and inside the rectangle with vertices 0,0,12,0,0,5, and 12,5. Find the maximum and minimum values of the objective function Q over the feasible region. Q=10x12yIn Problem 1-8, if necessary, review Theorem 1. In Problems 1-4, the feasible region is the set of points on and inside the rectangle with vertices 0,0,12,0,0,5, and 12,5. Find the maximum and minimum values of the objective function Q over the feasible region. Q=9x+20yIn Problems 1-8, if necessary, review Theorem 1. In Problems 5-8, the feasible region is the set of points on and inside the triangle with vertices 0,0,8,0 and 0,10. Find the maximum and minimum values of the objective function Q over the feasible region. Q=4x3yIn Problems 1-8, if necessary, review Theorem 1. In Problems 5-8, the feasible region is the set of points on and inside the triangle with vertices 0,0,8,0 and 0,10. Find the maximum and minimum values of the objective function Q over the feasible region. Q=3x+2yIn Problems 1-8, if necessary, review Theorem 1. In Problems 5-8, the feasible region is the set of points on and inside the triangle with vertices 0,0,8,0 and 0,10. Find the maximum and minimum values of the objective function Q over the feasible region. Q=6x+4yIn Problems 1-8, if necessary, review Theorem 1. In Problems 5-8, the feasible region is the set of points on and inside the triangle with vertices 0,0,8,0 and 0,10. Find the maximum and minimum values of the objective function Q over the feasible region. Q=10x8yIn Problems 9-12, graph the constant-profit lines through 3,3 and 6,6. Use a straightedge to identify the corner point where the maximum profit occurs (see Explore and Discuss 1). Confirm your answer by constructing a corner-point table. P=x+yIn Problems 9-12, graph the constant-profit lines through 3,3 and 6,6. Use a straightedge to identify the corner point where the maximum profit occurs (see Explore and Discuss 1). Confirm your answer by constructing a corner-point table. P=4x+yIn Problems 9-12, graph the constant-profit lines through 3,3 and 6,6. Use a straightedge to identify the corner point where the maximum profit occurs (see Explore and Discuss 1). Confirm your answer by constructing a corner-point table. P=3x+7yIn Problems 9-12, graph the constant-profit lines through 3,3 and 6,6. Use a straightedge to identify the corner point where the maximum profit occurs (see Explore and Discuss 1). Confirm your answer by constructing a corner-point table. P=9x+3yIn Problems 13-16, graph the constant-cost lines through 9,9 and 12,12. Use a straightedge to identify the corner point where the minimum cost occurs. Confirm your answer by constructing a corner-point table. C=7x+4yIn Problems 13-16, graph the constant-cost lines through 9,9 and 12,12. Use a straightedge to identify the corner point where the minimum cost occurs. Confirm your answer by constructing a corner-point table. C=7x+9yIn Problems 13-16, graph the constant-cost lines through 9,9 and 12,12. Use a straightedge to identify the corner point where the minimum cost occurs. Confirm your answer by constructing a corner-point table. C=3x+8yIn Problems 13-16, graph the constant-cost lines through 9,9 and 12,12. Use a straightedge to identify the corner point where the minimum cost occurs. Confirm your answer by constructing a corner-point table. C=2x+11ySolve the linear programming problems stated in Problems 17-38. MaximizeP=10x+75ysubjecttox+8y24x,y0Solve the linear programming problems stated in Problems 17-38. MaximizeP=30x+12ysubjectto3x+y18x,y0Solve the linear programming problems stated in Problems 17-38. MinimizeC=8x+9ysubjectto5x+6y60x,y0Solve the linear programming problems stated in Problems 17-38. MinimizeC=15x+25ysubjectto4x+7y28x,y0Solve the linear programming problems stated in Problems 17-38. MaximizeP=5x+5ysubjectto2x+y10x+2y8x,y0Solve the linear programming problems stated in Problems 17-38. MaximizeP=3x+2ysubjectto6x+3y243x+6y30x,y0Solve the linear programming problems stated in Problems 17-38. Minimizeandmaximizez=2x+3ysubjectto2x+y10x+2y8x,y0Solve the linear programming problems stated in Problems 17-38. Minimizeandmaximizez=8x+7ysubjectto4x+3y243x+4y8x,y0Solve the linear programming problems stated in Problems 17-38. MaximizeP=30x+40ysubjectto2x+y10x+y7x+2y12x,y0Solve the linear programming problems stated in Problems 17-38. MaximizeP=20x+10ysubjectto3x+y21x+y9x+3y21x,y0Solve the linear programming problems stated in Problems 17-38. Minimizeandmaximizez=10x+30ysubjectto2x+y16x+y12x+2y14x,y0Solve the linear programming problems stated in Problems 17-38. Minimizeandmaximizez=400x+100ysubjectto3x+y24x+y16x+3y30x,y0Solve the linear programming problems stated in Problems 17-38. MinimizeandmaximizeP=30x+10ysubjectto2x+2y46x+4y362x+y10x,y0Solve the linear programming problems stated in Problems 17-38. MinimizeandmaximizeP=2x+ysubjecttox+y26x+4y364x+2y20x,y0Solve the linear programming problems stated in Problems 17-38. MinimizeandmaximizeP=3x+5ysubjecttox+2y6x+y42x+3y12x,y0Solve the linear programming problems stated in Problems 17-38. MinimizeandmaximizeP=x+3ysubjectto2xy4x+2y4y6x,y0Solve the linear programming problems stated in Problems 17-38. MinimizeandmaximizeP=20x+10ysubjectto2x+3y302x+y262x+5y34x,y0Solve the linear programming problems stated in Problems 17-38. MinimizeandmaximizeP=12x+14ysubjectto2x+y6x+y153xy0x,y0Solve the linear programming problems stated in Problems 17-38. MaximizeP=20x+30ysubjectto0.6x+1.2y9600.03x+0.04y360.3x+0.2y270x,y0Solve the linear programming problems stated in Problems 17-38. MinimizeC=30x+10ysubjectto1.8x+0.9y2700.3x+0.2y540.01x+0.03y3.9x,y0Solve the linear programming problems stated in Problems 17-38. MaximizeP=525x+478ysubjectto275x+322y3,381350x+340y3,762425x+306y4,114x,y0Solve the linear programming problems stated in Problems 17-38. MaximizeP=300x+460ysubjectto245x+452y4,181290x+379y3,888390x+299y4,407x,y0In Problems 39 and 40, explain why Theorem 2 cannot be used to conclude that a maximum or minimum value exists. Graph the feasible regions and use graphs of the objective function z=xy for various values of z to discuss the existence of a maximum value and a minimum value. Minimizeandmaximizez=xysubjecttox2y02xy6x,y0In Problems 39 and 40, explain why Theorem 2 cannot be used to conclude that a maximum or minimum value exists. Graph the feasible regions and use graphs of the objective function z=xy for various values of z to discuss the existence of a maximum value and a minimum value. Minimizeandmaximizez=xysubjecttox2y62xy0x,y0Problems 41-48 refer to the bounded feasible region with corner points O=0,0,A=0,5,B=4,3, and C=5,0 that is determined by the system of inequalities x+2y103x+y15x,y0 If P=ax+10y, find all numbers a such that the maximum value of P occurs only at B.Problems 41-48 refer to the bounded feasible region with corner points O=0,0,A=0,5,B=4,3, and C=5,0 that is determined by the system of inequalities x+2y103x+y15x,y0 If P=ax+10y, find all numbers a such that the maximum value of P occurs only at A.Problems 41-48 refer to the bounded feasible region with corner points O=0,0,A=0,5,B=4,3, and C=5,0 that is determined by the system of inequalities x+2y103x+y15x,y0 If P=ax+10y, find all numbers a such that the maximum value of P occurs only at C.Problems 41-48 refer to the bounded feasible region with corner points O=0,0,A=0,5,B=4,3, and C=5,0 that is determined by the system of inequalities x+2y103x+y15x,y0 If P=ax+10y, find all numbers a such that the maximum value of P occurs at both A and B.Problems 41-48 refer to the bounded feasible region with corner points O=0,0,A=0,5,B=4,3, and C=5,0 that is determined by the system of inequalities x+2y103x+y15x,y0 If P=ax+10y, find all numbers a such that the maximum value of P occurs at both B and C.Problems 41-48 refer to the bounded feasible region with corner points O=0,0,A=0,5,B=4,3, and C=5,0 that is determined by the system of inequalities x+2y103x+y15x,y0 If P=ax+10y, find all numbers a such that the minimum value of P occurs at only C.Problems 41-48 refer to the bounded feasible region with corner points O=0,0,A=0,5,B=4,3, and C=5,0 that is determined by the system of inequalities x+2y103x+y15x,y0 If P=ax+10y, find all numbers a such that the minimum value of P occurs at both O and C.Problems 41-48 refer to the bounded feasible region with corner points O=0,0,A=0,5,B=4,3, and C=5,0 that is determined by the system of inequalities x+2y103x+y15x,y0 If P=ax+10y, explain why the minimum value of P cannot occur at B.In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Water skis. A manufacturing company makes two types of water-skis––a trick ski and slalom ski. The relevant manufacturing data are given in the table below. Labor-HoursperSkiDepartmentTrickSkiSlalomSkiMaximumLabor-HoursAvailableperDayFabricating64108Finishing1124 (A) If the profit on a trick ski is $40 and the profit on a slalom ski is $30, how many of each type of ski should be manufactured each day to realize a maximum profit? What is the maximum profit? (B) Discuss the effect on the production schedule and the maximum profit if the profit on a slalom ski decreases to $25. (C) Discuss the effect on the production schedule and the maximum profit if the profit on a slalom ski increases to $45.In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Furniture. A furniture manufacturing company manufactures dining-room tables and chairs. The relevant manufacturing data are given in the table below. Labor-HoursperUnitDepartmentTableChairMaximumLabor-HoursAvailableperDayAssembly82400Finishing21120Profitperunit$90$25 (A) How many tables and chairs should be manufactured each day to realize a maximum profit? What is the maximum profit? (B) Discuss the effect of production schedule and the maximum profit if the marketing department of the company decides that the number of chairs produced should be at least four times the number of tables produced.In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Production scheduling. A furniture company has two plants that produce the lumber used in manufacturing tables and chairs. In 1 day of operation, plant A can produce the lumber required to manufacture 20 tables and 60 chairs, and plant B can produce the lumber required to manufacture 25 tables and 50 chairs. The company needs enough lumber to manufacture at least 200 tables and 500 chairs. (A) If it costs $1,000 to operate plant A for 1 day and $900 to operate plant B for 1 day, how many days should each plant be operated to produce a sufficient amount of lumber at a minimum cost? What is the minimum cost? (B) Discuss the effect on the operating schedule and the minimum cost if the daily cost of operating plant A is reduced to $600 and all other data in part (A) remain the same. (C) Discuss the effect on the operating schedule and the minimum cost if the daily cost of operating plant B is reduced to $800 and all other data in part (A) remain the same.In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Computers. An electronics firm manufactures two types of personal computers––a standard model and a portable model. The production of a standard computer requires a capital expenditure of $400 and 40 hours of labor. The production of a portable computer requires a capital expenditure of $250 and 30 hours of labor. The firm has $20,000 capital and 2,160 labor-hours available for production of standard and portable computers. (A) What is the maximum number of computers the company is capable of producing? (B) If each standard computer contributes a profit of $320 and each portable model contributes a profit of $220, how much profit will the company make by producing the maximum number of computers determined in part (A)? Is this the maximum profit? If not, what is the maximum profit?In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Transportation. The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 40 students, requires 3 chaperones, and costs $1,200 to rent. Each van can transport 8 students, requires 1 chaperone, and costs $100 to rent. Since there are 400 students in the senior class that may be eligible to go on the trip, the officers must plan to accommodate at least 400 students. Since only 36 parents have volunteered to serve as chaperones, the officers must plan to sue at most 36 chaperones. How many vehicles of each type should the officers rent in order to minimize the transportation costs? What are the minimal transportation costs?In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Refer to Problem 53. If each van can transport 7 people and there are 35 available chaperones, show that the optimal solution found graphically involves decimals. Find all feasible solutions with integer coordinates and identify the one that minimizes the transportation costs. Can this optimal integer solution be obtained by rounding the optimal decimal solution? Explain.In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Investment. An investor has $60,000 to invest in a CD and a mutual fund. The CD yields 5% and the mutual fund yields an average of 9%. The mutual fund requires a minimum investment of $10,000, and the investor requires that at least twice as much should be invested in CDs as in the mutual fund. How much should be invested in CDs and how much in the mutual fund to maximize the return? What is the maximum return?In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Investment. An investor has $24,000 to invest in bonds of AAA and B qualities. The AAA bonds yield an average of 6% and the B bonds yield 10%. The investor requires that at least three times as much money should be invested in AAA bonds as in B bonds. How much should be invested in each type of bond to maximize the return? What is the maximum return?In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Pollution Control. Because of new federal regulations on pollution, a chemical plant introduced a new, more expensive process to supplement or replace an older process used in the production of a particular chemical. The older process emitted 20 grams of sulfur dioxide and 40 grams of particulate matter into the atmosphere for each gallon of chemical produced. The new process emits 5 grams of sulfur dioxide and 20 grams of particulate matter for each gallon produced. The company makes a profit of 60c per gallon and 20c per gallon on the old and new process, respectively. (A) If the government allows the plant to emit no more than 16,000 grams of sulfur dioxide and 30,000 grams of particulate matter daily, how many gallons of the chemical should be produced by each process to maximize daily profit? What is the maximum daily profit? (B) Discuss the effect on the production schedule and the maximum profit if the government decides to restrict emissions of sulfur dioxide to 11,500 grams daily and all other data remain unchanged. (C) Discuss the effect on the production schedule and the maximum profit if the government decides to restrict emissions of sulfur dioxide to 7,200 grams daily and all other data remain unchanged.In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Capital expansion. A fast-food chain plans to expand by opening several new restaurants. The chain operates two types of restaurants, drive-through and full-service. A drive-through restaurant costs $100,000 to construct, requires 5 employees, and has an expected annual revenue of $200,000. A full-service restaurant costs $150,000 to construct, requires 15 employees, and has an expected annual revenue of $500,000. The chain has $2,400,000 in capital available for expansion. Labor contracts require that they hire no more than 210 employees, and licensing restrictions require that they open no more than 20 new restaurants. How many restaurants of each type should the chain open in order to maximize the expected revenue? What is the maximum expected revenue? How much of their capital will they use and how many employees will they hire?In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Fertilizer. A fruit grower can use two types of fertilizer in his orange grove, brand A and brand B. The amounts (in pounds) of nitrogen, phosphoric acid, and chloride in a bag of each brand are given in the table. Tests indicate that the grove needs at least 1,000 pounds of phosphoric acid and at most 400 pounds of chloride. (A) If the grower wants to maximize the amount of nitrogen added to the grove, how many bags of each mix should be used? How much nitrogen will be added? (B) If the grower wants to minimize the amount of nitrogen added to the grove, how many bags of each mix should be used? How much nitrogen will be added?In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Nutrition. A dietician is to arrange a special diet composed of two foods, M and N. Each ounce of food M contains 30 units of calcium, 10 units of iron, 10 units of vitamin A, and 8 units of cholesterol. Each ounce of food N contains 10 units of calcium, 10 units of iron, 30 units of vitamin A, and 4 units of cholesterol. If the minimum daily requirements are 360 units of calcium, 160 units of iron, and 240 units of vitamin A, how many ounces of each food should be used to meet the minimum requirements and at the same time minimize the cholesterol intake? What is the minimum cholesterol intake?In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Plant food. A farmer can buy two types of plant food, mix A and mix B. Each cubic yard of mix A contains 20 pounds of phosphoric acid, 30 pounds of nitrogen, and 5 pounds of potash. Each cubic yard of mix B contains 10 pounds of phosphoric acid, 30 pounds of nitrogen, and 10 pounds of potash. The minimum monthly requirements are 460 pounds of phosphoric acid, 960 pounds of nitrogen, and 220 pounds of potash. If mix A costs $30 per cubic yard and mix B costs $35 per cubic yard, how many cubic yards of each mix should the farmer blend to meet the minimum monthly requirements at a minimal cost? What is this cost?In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Animal food. A laboratory technician in a medical research center is asked to formulate a diet from two commercially packaged foods, food A and food B, for a group of animals. Each ounce of food A contains 8 units of fat, 16 units of carbohydrate, and 2 units of protein. Each ounce of food B contains 4 units of fat, 32 units of carbohydrate, and 8 units of protein. The minimum daily requirements are 176 units of fat, 1,024 units of carbohydrate, and 384 units of protein. If food A costs 5c per ounce and food B costs 5c per ounce, how many ounces of each food should be used to meet the minimum daily requirements at the least cost? What is the cost for this amount of food?In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Psychology. A psychologist uses two types of boxes with mice and rats. The amounts of time (in minutes) that each mouse and each rat spends in each box per day is given in the table. What is the maximum number of mice and rats that can be used in this experiment? How many mice and how many rats produce this maximum?In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method. Sociology. A city council voted to conduct a study on inner-city community problems using sociologists and research assistants from a nearby university. Allocation of time and costs per week are given in the table. How many sociologists and how many research assistants should be hired to minimize the cost and meet the weekly labor-hour requirements? What is the minimum weekly cost?Graph each inequality. x2y3Graph each inequality. 3y5x30Graph the systems in Problems 3-6 and indicate whether each solution region is bounded or unbounded. Find the coordinates of each corner point. 5x+9y90x,y0Graph the systems in Problems 3-6 and indicate whether each solution region is bounded or unbounded. Find the coordinates of each corner point. 15x+16y1,200x,y0Graph the systems in Problems 3-6 and indicate whether each solution region is bounded or unbounded. Find the coordinates of each corner point. 2x+y83x+9y27x,y0Graph the systems in Problems 3-6 and indicate whether each solution region is bounded or unbounded. Find the coordinates of each corner point. 3x+y92x+4y16x,y0In Exercises 7 and 8, state the linear inequality whose graph is given in the figure. Write the boundary line equation in the form Ax+By=C, with A,B, and C integers, before stating the inequality.In Exercises 7 and 8, state the linear inequality whose graph is given in the figure. Write the boundary line equation in the form Ax+By=C, with A,B, and C integers, before stating the inequality.Solve the linear programming problems in Problems 9-13. MaximizeP=2x+6ysubjecttox+2y82x+y10x,y0Solve the linear programming problems in Problems 9-13. MinimizeC=5x+2ysubjecttox+3y152x+y20x,y0Solve the linear programming problems in Problems 9-13. MaximizeP=3x+4ysubjecttox+2y12x+y72x+y10x,y0Solve the linear programming problems in Problems 9-13. MinimizeC=8x+3ysubjecttox+y102x+y15x3x,y0Solve the linear programming problems in Problems 9-13. MaximizeP=3x+2ysubjectto2x+y22x+3y26x10y10x,y0Electronics. A company uses two machines to solder circuit boards, an oven and a wave soldering machine. A circuit board for a calculator needs 4 minutes in the oven and 2 minutes on the wave machine, while a circuit board for a toaster requires 3 minutes in the oven and 1 minute on the wave machine. (Source: Universal Electronics) (A) How many circuit boards for calculators and toasters can be produced if the oven is available for 5 hours? Express your answer as a linear inequality with appropriate non-negative restrictions and draw its graph. (B) How many circuit boards for calculators and toasters can be produced if the wave machine is available for 2 hours? Express your answer as a linear inequality with appropriate nonnegative restrictions and draw its graph.In problems 15 and 16, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve the problem by the indicated method. Sail manufacture. South Shore Sail Loft manufactures regular and competition sails. Each regular sail takes 2 hours to cut and 4 hours to sew. Each competition sail takes 3 hours to cut and 10 hours to sew. There are 150 hours available in the cutting department and 380 hours available in the sewing department. (A) If the Loft makes a profit of $100 on each regular sail and $200 on each competition sail, how many sails of each type should the company manufacture to maximize its profit? What is the maximum profit? (B) An increase in the demand for competition sails causes the profit on a competition sail to rise to $260. Discuss the effect of this change on the number of sails manufactured and on the maximum profit. (C) An decrease in the demand for competition sails causes the profit on a competition sail to drop to $140. Discuss the effect of this change on the number of sails manufactured and on the maximum profit.In problems 15 and 16, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve the problem by the indicated method. Animal food. A special diet for laboratory animals is to contain at least 850 units of vitamins, 800 units of minerals, and 1,150 calories. There are two feed mixes available, mix A and mix B. A gram of mix A contains 2 units of vitamins, 2 units of minerals, and 4 calories. A gram of mix B contains 5 units of vitamins, 4 units of minerals, and 5 calories. (A) If mix A costs $0.04 per gram and mix B costs $0.09 per gram, how many grams of each mix should be used to satisfy the requirements of the diet at minimal cost? What is the minimum cost? (B) If the price of mix B decreases to $0.06 per gram, discuss the effect of this change on the solution in part (A). (C) If the price of mix B increases to $0.12 per gram, discuss the effect of this change on the solution in part (A).The following linear programming problem has only one problem constraint: MaximizeP=2x1+3x2subjectto4x1+5x220x1,x20 Solve it by the table method, then solve it by graphing, and compare the two solutions.Use the table method to solve the following linear programming problem, and explain why one of the rows in the table cannot be completed to a basic solution: MaximizeP=10x1+12x2 Subjecttox1+x22x1+x23x1,x20Use the table method to solve the following linear programming problem, and explain why one of the rows in the table cannot be completed to a basic solution: MaximizeP=10x1+12x2subjecttox1+x22x1+x23x1,x20Refer to Example 1. Find the basic solution for which x2=0 and s1=0.Construct the table of basic solutions and use it to solve the following linear programming problem: MaximizeP=30x1+40x2subjectto2x1+3x2244x1+3x236x1,x20Construct the table of basic solutions and use it to solve the following linear programming problem: MaximizeP=36x1+24x2subjecttox1+2x28x1+x252x1+x28x1,x20Refer to Table 5. For the basic solution x1,x2,s1,s2,s3=27,6,9,0,0 of row 10, classify the variables as basic or nonbasic.In Problems 1-8, evaluate the expression. (If necessary, review Section B.3). 8!3!5!In Problems 1-8, evaluate the expression. (If necessary, review Section B.3). 10!2!8!In Problems 1-8, evaluate the expression. (If necessary, review Section B.3). 11!9!2!In Problems 1-8, evaluate the expression. (If necessary, review Section B.3). 7!4!3!In Problems 1-8, evaluate the expression. (If necessary, review Section B.3). In how many ways can two variables be chosen from x1,x2,s1,s2,s3,s4,s5 and assigned the value 0 ?In Problems 1-8, evaluate the expression. (If necessary, review Section B.3). In how many ways can three variables be chosen from x1,x2,x3,s1,s2,s3 and assigned the value 0 ?In Problems 1-8, evaluate the expression. (If necessary, review Section B.3). In how many ways can four variables be chosen from x1,x2,x3,x4,s1,s2,s3,s4 and assigned the value 0 ?In Problems 1-8, evaluate the expression. (If necessary, review Section B.3). In how many ways can three variables be chosen from x1,x2,s1,s2,s3,s4,s5,s6 and assigned the value 0 ?Problems 9-12 refer to the system 2x1+5x2+s1=10x1+3x2+s2=8 Find the solution of the system for which x1=0,s1=0.Problems 9-12 refer to the system 2x1+5x2+s1=10x1+3x2+s2=8 Find the solution of the system for which x1=0,s2=0.Problems 9-12 refer to the system 2x1+5x2+s1=10x1+3x2+s2=8 Find the solution of the system for which x2=0,s2=0.Problems 9-12 refer to the system 2x1+5x2+s1=10x1+3x2+s2=8 Find the solution of the system for which x2=0,s1=0.In Problems 13-20, write the e-system obtained via slack variables for the given linear programming problem. MaximizeP=5x1+7x2subjectto2x1+3x296x1+7x213x1,x20In Problems 13-20, write the e-system obtained via slack variables for the given linear programming problem. MaximizeP=35x1+25x2subjectto10x1+15x21005x1+20x2120x1,x20In Problems 13-20, write the e-system obtained via slack variables for the given linear programming problem. MaximizeP=3x1+5x2subjectto12x114x25519x1+5x2408x1+11x264x1,x20In Problems 13-20, write the e-system obtained via slack variables for the given linear programming problem. MaximizeP=13x1+25x2subjectto3x1+5x2278x1+3x2194x1+9x234x1,x20In Problems 13-20, write the e-system obtained via slack variables for the given linear programming problem. MaximizeP=4x1+7x2subjectto6x1+5x218x1,x20In Problems 13-20, write the e-system obtained via slack variables for the given linear programming problem. MaximizeP=13x1+8x2subjecttox1+2x220x1,x20In Problems 13-20, write the e-system obtained via slack variables for the given linear programming problem. MaximizeP=x1+2x2subjectto4x13x2125x1+2x2253x1+7x2322x1+x29x1,x20In Problems 13-20, write the e-system obtained via slack variables for the given linear programming problem. MaximizeP=8x1+9x2subjectto30x125x27510x1+13x2305x1+18x24040x1+36x285x1,x20Problems 21-30 refer to the table below of the six basic solutions to the e-system 2x1+3x2+s1=244x1+3x2+s2=36x1x2s1s2A002436B08012C012120D120012E9060F6400 In basic solution A, which variables are basic?Problems 21-30 refer to the table below of the six basic solutions to the e-system 2x1+3x2+s1=244x1+3x2+s2=36x1x2s1s2A002436B08012C012120D120012E9060F6400 In basic solution B, which variables are nonbasic?Problems 21-30 refer to the table below of the six basic solutions to the e-system 2x1+3x2+s1=244x1+3x2+s2=36x1x2s1s2A002436B08012C012120D120012E9060F6400 In basic solution C, which variables are nonbasic?Problems 21-30 refer to the table below of the six basic solutions to the e-system 2x1+3x2+s1=244x1+3x2+s2=36x1x2s1s2A002436B08012C012120D120012E9060F6400 In basic solution D, which variables are basic?Problems 21-30 refer to the table below of the six basic solutions to the e-system 2x1+3x2+s1=244x1+3x2+s2=36x1x2s1s2A002436B08012C012120D120012E9060F6400 Which of the six basic solutions are feasible? Explain.Problems 21-30 refer to the table below of the six basic solutions to the e-system 2x1+3x2+s1=244x1+3x2+s2=36x1x2s1s2A002436B08012C012120D120012E9060F6400 Which of the basic solutions are not feasible? Explain.Problems 21-30 refer to the table below of the six basic solutions to the e-system 2x1+3x2+s1=244x1+3x2+s2=36x1x2s1s2A002436B08012C012120D120012E9060F6400 Use the basic feasible solutions to minimize P=2x1+5x2.Problems 21-30 refer to the table below of the six basic solutions to the e-system 2x1+3x2+s1=244x1+3x2+s2=36x1x2s1s2A002436B08012C012120D120012E9060F6400 Use the basic feasible solutions to maximize P=8x1+5x2.Problems 21-30 refer to the table below of the six basic solutions to the e-system 2x1+3x2+s1=244x1+3x2+s2=36x1x2s1s2A002436B08012C012120D120012E9060F6400 Describe geometrically the set of all points in the plane such that s10.Problems 21-30 refer to the table below of the six basic solutions to the e-system 2x1+3x2+s1=244x1+3x2+s2=36x1x2s1s2A002436B08012C012120D120012E9060F6400 Describe geometrically the set of all points in the plane such that s20.Problems 31-40 refer to the partially completed table below of the 10 basic solutions to the e-system x1+x2+s1=242x1+x2+s2=304x1+x2+s3=48x1x2s1s2s3A00243048B0240624C0306018D04824180E24001848F1509012G00H00I00J00 In the basic solution C, which variables are basic?Problems 31-40 refer to the partially completed table below of the 10 basic solutions to the e-system x1+x2+s1=242x1+x2+s2=304x1+x2+s3=48x1x2s1s2s3A00243048B0240624C0306018D04824180E24001848F1509012G00H00I00J00 In the basic solution E, which variables are nonbasic?Problems 31-40 refer to the partially completed table below of the 10 basic solutions to the e-system x1+x2+s1=242x1+x2+s2=304x1+x2+s3=48x1x2s1s2s3A00243048B0240624C0306018D04824180E24001848F1509012G00H00I00J00 In the basic solution G, which variables are nonbasic?Problems 31-40 refer to the partially completed table below of the 10 basic solutions to the e-system x1+x2+s1=242x1+x2+s2=304x1+x2+s3=48x1x2s1s2s3A00243048B0240624C0306018D04824180E24001848F1509012G00H00I00J00 In the basic solution I, which variables are basic?Problems 31-40 refer to the partially completed table below of the 10 basic solutions to the e-system x1+x2+s1=242x1+x2+s2=304x1+x2+s3=48x1x2x3x4x5A00243048B0240624C0306018D04824180E24001848F1509012G00H00I00J00 Which of the basic solutions A through F are not feasible? Explain.Problems 31-40 refer to the partially completed table below of the 10 basic solutions to the e-system x1+x2+s1=242x1+x2+s2=304x1+x2+s3=48x1x2x3x4x5A00243048B0240624C0306018D04824180E24001848F1509012G00H00I00J00 Which of the basic solutions A through F are feasible? Explain.Problems 31-40 refer to the partially completed table below of the 10 basic solutions to the e-system x1+x2+s1=242x1+x2+s2=304x1+x2+s3=48x1x2x3x4x5A00243048B0240624C0306018D04824180E24001848F1509012G00H00I00J00 Find basic solution G.Problems 31-40 refer to the partially completed table below of the 10 basic solutions to the e-system x1+x2+s1=242x1+x2+s2=304x1+x2+s3=48x1x2s1s2s3A00243048B0240624C0306018D04824180E24001848F1509012G00H00I00J00 Find basic solution H.Problems 31-40 refer to the partially completed table below of the 10 basic solutions to the e-system x1+x2+s1=242x1+x2+s2=304x1+x2+s3=48x1x2s1s2s3A00243048B0240624C0306018D04824180E24001848F1509012G00H00I00J00 Find basic solution I.Problems 31-40 refer to the partially completed table below of the 10 basic solutions to the e-system x1+x2+s1=242x1+x2+s2=304x1+x2+s3=48x1x2s1s2s3A00243048B0240624C0306018D04824180E24001848F1509012G00H00I00J00 Find basic solution J.In Problems 41-48, convert the given i-system to an e-system using slack variables. Then construct a table of all basic solutions of the e-system. For each basic solution, indicate whether or not it is feasible. 4x1+5x220x1,x20In Problems 41-48, convert the given i-system to an e-system using slack variables. Then construct a table of all basic solutions of the e-system. For each basic solution, indicate whether or not it is feasible. 3x1+8x224x1,x20In Problems 41-48, convert the given i-system to an e-system using slack variables. Then construct a table of all basic solutions of the e-system. For each basic solution, indicate whether or not it is feasible. x1+x26x1+4x212x1,x20In Problems 41-48, convert the given i-system to an e-system using slack variables. Then construct a table of all basic solutions of the e-system. For each basic solution, indicate whether or not it is feasible. 5x1+x215x1+x27x1,x20In Problems 41-48, convert the given i-system to an e-system using slack variables. Then construct a table of all basic solutions of the e-system. For each basic solution, indicate whether or not it is feasible. 2x1+5x220x1+2x29x1,x20In Problems 41-48, convert the given i-system to an e-system using slack variables. Then construct a table of all basic solutions of the e-system. For each basic solution, indicate whether or not it is feasible. x1+3x2185x1+4x235x1,x20In Problems 41-48, convert the given i-system to an e-system using slack variables. Then construct a table of all basic solutions of the e-system. For each basic solution, indicate whether or not it is feasible. x1+2x224x1+x2152x1+x224x1,x20In Problems 41-48, convert the given i-system to an e-system using slack variables. Then construct a table of all basic solutions of the e-system. For each basic solution, indicate whether or not it is feasible. 5x1+4x22405x1+2x21505x1+x2120x1,x20In Problems 49-54, graph the system of inequalities from the given problem, and list the corner points of the feasible region. Verify that the corner points of the feasible region correspond to the basic feasible solutions of the associated e-system. Problem 41In Problems 49-54, graph the system of inequalities from the given problem, and list the corner points of the feasible region. Verify that the corner points of the feasible region correspond to the basic feasible solutions of the associated e-system. Problem 42In Problems 49-54, graph the system of inequalities from the given problem, and list the corner points of the feasible region. Verify that the corner points of the feasible region correspond to the basic feasible solutions of the associated e-system. Problem 43In Problems 49-54, graph the system of inequalities from the given problem, and list the corner points of the feasible region. Verify that the corner points of the feasible region correspond to the basic feasible solutions of the associated e-system. Problem 44In Problems 49-54, graph the system of inequalities from the given problem, and list the corner points of the feasible region. Verify that the corner points of the feasible region correspond to the basic feasible solutions of the associated e-system. Problem 45In Problems 49-54, graph the system of inequalities from the given problem, and list the corner points of the feasible region. Verify that the corner points of the feasible region correspond to the basic feasible solutions of the associated e-system. Problem 46For a standard maximization problem in standard form, with two decision variables x1 and x2, explain why the feasible region is not empty.For a standard maximization problem in standard form, with k decision variables, x1,x2,......,xk, explain why the feasible region is not empty.If 5x1+4x21,000 is one of the problem constraints in a standard maximization problem in standard form with two decision variables, explain why the optimal value of the objective function exists. [Hint: See Theorem 2 in Section 5.3].If a1x1+a2x2b is one of the problem constraints in a standard maximization problem in standard form with two decision variables, and a1 and a2 are both positive, explain why the optimal value of the objective function exists. [Hint: See Theorem 2 in Section 5.3].In Problems 59-66, solve the given linear programming problem using the table method (the table of basic solutions was constructed in Problems 41-48) MaximizeP=10x1+9x2subjectto4x1+5x220x1,x20In Problems 59-66, solve the given linear programming problem using the table method (the table of basic solutions was constructed in Problems 41-48) MaximizeP=4x1+7x2subjectto3x1+8x224x1,x20In Problems 59-66, solve the given linear programming problem using the table method (the table of basic solutions was constructed in Problems 41-48) MaximizeP=15x1+20x2subjecttox1+x26x1+4x212x1,x20In Problems 59-66, solve the given linear programming problem using the table method (the table of basic solutions was constructed in Problems 41-48) MaximizeP=5x1+20x2subjectto5x1+x215x1+x27x1,x20In Problems 59-66, solve the given linear programming problem using the table method (the table of basic solutions was constructed in Problems 41-48) MaximizeP=25x1+10x2subjectto2x1+5x220x1+2x29x1,x20In Problems 59-66, solve the given linear programming problem using the table method (the table of basic solutions was constructed in Problems 41-48) MaximizeP=40x1+50x2subjecttox1+3x2185x1+4x235x1,x20In Problems 59-66, solve the given linear programming problem using the table method (the table of basic solutions was constructed in Problems 41-48) MaximizeP=30x1+40x2subjecttox1+2x224x1+x2152x1+x224x1,x20In Problems 59-66, solve the given linear programming problem using the table method (the table of basic solutions was constructed in Problems 41-48) MaximizeP=x1+x2subjectto5x1+4x22405x1+2x21505x1+x2120x1,x20In Problems 67-70, explain why the linear programming problem has no optimal solution. MaximizeP=8x1+9x2subjectto3x17x242x1,x20In Problems 67-70, explain why the linear programming problem has no optimal solution. MaximizeP=12x1+8x2subjectto2x1+10x230x1,x20In Problems 67-70, explain why the linear programming problem has no optimal solution. MaximizeP=6x1+13x2subjectto4x1+x244x15x212x1,x20In Problems 67-70, explain why the linear programming problem has no optimal solution. MaximizeP=18x1+11x2subjectto3x12x263x1+2x26x1,x20In Problems 71-72, explain why the linear programming problem has an optimal solution, and find it using the table method. MaximizeP=20x1+25x2subjectto2x1+x250x1100x1,x20In Problems 71-72, explain why the linear programming problem has an optimal solution, and find it using the table method. MaximizeP=15x1+12x2subjectto2x1+5x2102x1x26x1,x20A linear programming problem has four decision variables x1,x2,x3,x4, and six problem constraints. How many rows are there in the table of basic solutions of the associated e-system?A linear programming problem has five decision variables x1,x2,x3,x4,x5 and six problem constraints. How many rows are there in the table of basic solutions of the associated e-system?A linear programming problem has 30 decision variables x1,x2,.......,x30 and 42 problem constraints. How many rows are there in the table of basic solutions of the associated e-system? (Write the answer using scientific notation.)A linear programming problem has 40 decision variables x1,x2,.......,x40 and 85 problem constraints. How many rows are there in the table of basic solutions of the associated e-system? (Write the answer using scientific notation.)Graph the feasible region for the linear programming problem in Example 1 and trace the path to the optimal solution determined by the simplex method.Solve the following linear programming problem using the simplex method: MaximizeP=2x1+3x2subjectto5x1+x29x1+x25x1,x20Solve using the simplex method: MaximizeP=2x1+3x2subjectto3x1+4x212x22x1,x20Repeat Example 3 modified as follows:For the simplex tableau in Problems 1-4, (A) Identify the basic and nonbasic variables. (B) Find the corresponding basic feasible solution. (C) determine whether the optimal solution has been found, an additional pivot is required, or the problem has no optimal solution. x1x2s1s2P210301230120154004150For the simplex tableau in Problems 1-4, (A) Identify the basic and nonbasic variables. (B) Find the corresponding basic feasible solution. (C) determine whether the optimal solution has been found, an additional pivot is required, or the problem has no optimal solution. x1x2s1s2P142001002310250560135For the simplex tableau in Problems 1-4, (A) Identify the basic and nonbasic variables. (B) Find the corresponding basic feasible solution. (C) determine whether the optimal solution has been found, an additional pivot is required, or the problem has no optimal solution. x1x2x3s1s2s3P20131005010200015100411012400240145For the simplex tableau in Problems 1-4, (A) Identify the basic and nonbasic variables. (B) Find the corresponding basic feasible solution. (C) determine whether the optimal solution has been found, an additional pivot is required, or the problem has no optimal solution. x1x2x3s1s2s3P0211400501202102130050011054030127In Problems 5-8, find the pivot element, identify the entering and exiting variables, and perform one pivot operation. x1x2s1s2P1410043501024850010In Problems 5-8, find the pivot element, identify the entering and exiting variables, and perform one pivot operation. x1x2s1s2P1610036310105120010In Problems 5-8, find the pivot element, identify the entering and exiting variables, and perform one pivot operation. x1x2s1s2s3P2110004301100800201024030015In Problems 5-8, find the pivot element, identify the entering and exiting variables, and perform one pivot operation. x1x2s1s2s3P002110210401030150201100605118In Problems 9-12, (A) Using the slack variables, write the initial system for each linear programming problem. (B) Write the simplex tableau, circle the first pivot, and identify the entering and exiting variables. (C) Use the simplex method to solve the problem. MaximizeP=15x1+10x2subjectto2x1+x210x1+3x210x1,x20In Problems 9-12, (A) Using the slack variables, write the initial system for each linear programming problem. (B) Write the simplex tableau, circle the first pivot, and identify the entering and exiting variables. (C) Use the simplex method to solve the problem. MaximizeP=3x1+2x2subjectto5x1+2x2203x1+2x216x1,x20In Problems 9-12, (A) Using the slack variables, write the initial system for each linear programming problem. (B) Write the simplex tableau, circle the first pivot, and identify the entering and exiting variables. (C) Use the simplex method to solve the problem. Repeat Problem 9 with the objective function changed to P=30x1+x2.In Problems 9-12, (A) Using the slack variables, write the initial system for each linear programming problem. (B) Write the simplex tableau, circle the first pivot, and identify the entering and exiting variables. (C) Use the simplex method to solve the problem. Repeat Problem 10 with the objective function changed to P=x1+3x2.Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=30x1+40x2subjectto2x1+x210x1+x27x1+2x212x1,x20Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=15x1+20x2subjectto2x1+x29x1+x26x1+2x210x1,x20Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=2x1+3x2subjectto2x1+x22x1+x25x26x1,x20Solve the linear programming problems in Problems 13-32 using the simplex method. Repeat Problem 15 with P=x1+3x2.Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=x1+2x2subjecttox1+x22x1+3x212x14x24x1,x20Solve the linear programming problems in Problems 13-32 using the simplex method. Repeat Problem 17 with P=x1+2x2.Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=15x1+36x2subjecttox1+3x26x1,x20Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=8x1+7x2subjectto2x15x210x1,x20Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=27x1+64x2subjectto8x13x224x1,x20Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=45x1+30x2subjectto4x1+3x212x1,x20Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=5x1+2x2x3subjecttox1+x2x3102x1+4x2+3x330x1,x2,x30Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=4x13x2+2x3subjecttox1+2x2x353x1+2x2+2x322x1,x2,x30Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=2x1+3x2+4x3subjecttox1+x34x2+x33x1,x2,x30Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=x1+x2+2x3subjecttox12x2+x392x1+x2+2x328x1,x2,x30Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=4x1+3x2+2x3subjectto3x1+2x2+5x3232x1+x2+x38x1+x2+2x37x1,x2,x30Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=4x1+2x2+3x3subjecttox1+x2+x3112x1+3x2+x320x1+3x2+2x320x1,x2,x30Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=20x1+30x2subjectto0.6x1+1.2x29600.03x1+0.04x2360.3x1+0.2x2270x1,x20Solve the linear programming problems in Problems 13-32 using the simplex method. Repeat Problem 29 with P=20x1+20x2.Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=x1+2x2+3x3subjectto2x1+2x2+8x3600x1+3x2+2x36003x1+2x2+x3400x1,x2,x30Solve the linear programming problems in Problems 13-32 using the simplex method. MaximizeP=10x1+50x2+10x3subjectto3x1+3x2+3x3666x12x2+4x3483x1+6x2+9x3108x1,x2,x30In Problems 33 and 34, first solve the linear programming problem by the simplex method, keeping track of the basic feasible solutions at each step. Then graph the feasible region and illustrate the path to the optimal solution determined by the simplex method. MaximizeP=2x1+5x2subjecttox1+2x240x1+3x248x1+4x260x214x1,x20In Problems 33 and 34, first solve the linear programming problem by the simplex method, keeping track of the basic feasible solutions at each step. Then graph the feasible region and illustrate the path to the optimal solution determined by the simplex method. MaximizeP=5x1+3x2subjectto5x1+4x21002x1+x2284x1+x242x110x1,x20Solve Problems 35 and 36 by the simplex method and also by graphing (the geometric method). Compare and contrast the results. MaximizeP=2x1+3x2subjectto2x1+x24x210x1,x20Solve Problems 35 and 36 by the simplex method and also by graphing (the geometric method). Compare and contrast the results. MaximizeP=2x1+3x2subjecttox1+x22x24x1,x20In Problems 37-40, there is a tie for the choice of the first pivot column. Use the simplex method to solve each problem two different ways: first by choosing column 1 as the first pivot column, and then by choosing column 2 as the first pivot column. Discuss the relationship between these two solutions. MaximizeP=x1+x2subjectto2x1+x216x16x210x1,x20In Problems 37-40, there is a tie for the choice of the first pivot column. Use the simplex method to solve each problem two different ways: first by choosing column 1 as the first pivot column, and then by choosing column 2 as the first pivot column. Discuss the relationship between these two solutions. MaximizeP=x1+x2subjecttox1+2x210x16x24x1,x20In Problems 37-40, there is a tie for the choice of the first pivot column. Use the simplex method to solve each problem two different ways: first by choosing column 1 as the first pivot column, and then by choosing column 2 as the first pivot column. Discuss the relationship between these two solutions. MaximizeP=3x1+3x2+2x3subjecttox1+x2+2x3202x1+x2+4x332x1,x2,x30In Problems 37-40, there is a tie for the choice of the first pivot column. Use the simplex method to solve each problem two different ways: first by choosing column 1 as the first pivot column, and then by choosing column 2 as the first pivot column. Discuss the relationship between these two solutions. MaximizeP=2x1+2x2+x3subjecttox1+x2+3x3102x1+4x2+5x324x1,x2,x30In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Manufacturing: resource allocation. A small company manufactures three different electronic components for computers. Component A requires 2 hours of fabrication and 1 hour of assembly; component B requires 3 hours of fabrication and 1 hour of assembly; and component C requires 2 hours of fabrication and 2 hours of assembly. The company has up to 1,000 labor-hours of fabrication time and 800 labor-hours of assembly time available per week. The profit on each component, A,B, and C, is $7,$8, and $10, respectively. How many components of each type should the company manufacture each week in order to maximize the profit (assuming that all components manufactured can be sold)? What is the maximum profit?In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Manufacturing: resource allocation. Solve Problem 41 with the additional restriction that the combined total number of components produced each week cannot exceed 420. Discuss the effect of this restriction on the solution to Problem 41.In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Investment. An investor has at most $100,000 to invest in government bonds, mutual funds, and money market funds. The average yields for government bonds, mutual funds, and money market funds are 8%,13%, and 15%, respectively. The investor’s policy requires that the total amount invested in mutual and money market funds not exceed the amount invested in government bonds. How much should be invested in each type of investment in order to maximize the return? What is the maximum return?In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Investment. Repeat Problem 43 under the additional assumption that no more than $30,000 can be invested in money market funds.In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Advertising. A department store has up to $20,000 to spend on television advertising for a sale, All ads will be places with one television station. A 30 -second ad costs $1,000 on daytime TV and is viewed by 14,000 potential customers, $2,000 on prime-time TV and is viewed by 24,000 potential customers, and $1,500 on late-night TV and is viewed by 18,000 potential customers. The television station will not accept a total of more than 15 ads in all three time periods. How many ads should be placed in each time period in order to maximize the number of potential customers who will see the ads? How many potential customers will see the ads? (Ignore repeated viewings of the ad by the same potential customer.)In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Advertising. Repeat Problem 45 if the department store increases its budget to $24,000 and requires that at least half of the ads be placed during prime-time.In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Home construction. A contractor is planning a new housing development consisting of colonial, split-level, and ranch-style houses. A colonial house requires 12 acre of land, $60,000 capital, and 4,000 labor-hours to construct, and returns a profit of $20,000. A split-level house requires 12 acre of land, $60,000 capital, and 3,000 labor-hours to construct, and returns a profit of $18,000. A ranch house requires 1 acre of land, $80,000 capital, and 4,000 labor-hours to construct, and returns a profit of $24,000. The contractor has 30 acres of land, $3,200,000 capital, and 180,000 labor-hours available. How many houses of each type should be constructed to maximize the contractor’s profit? What is the maximum profit?In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Bicycle manufacturing. A company manufactures three-speed, five-speed and ten-speed bicycles. Each bicycle passes through three departments: fabrication, painting plating, and final assembly. The relevant manufacturing data are given in the table. Labor-HoursperBicycleThree-SpeedFive-SpeedTen-SpeedMamimumLabor-HoursAvailableperdayFabrication345120Paintingplating535130Finalassembly435120Profitperbicycle$8070100 How many bicycles of each type should the company manufacture per day in order to maximize its profit? What is the maximum profit?In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Home building. Repeat Problem 47 if the profit on a colonial house decreases from $20,000 to $17,000 and all other data remain the same. If the slack associated with any problem constraints is nonzero, find it.In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Bicycle manufacturing. Repeat Problem 48 if the profit on a ten-speed bicycle increases from $100 to $110 and all other data remain the same. If the slack associated with any problem constraint is nonzero, find it.In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Home building. Repeat Problem 47 if the profit on a colonial house increases from $20,000 to $25,000 and all other data remain the same. If the slack associated with any problem constraint is nonzero, find it.In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Bicycle manufacturing. Repeat Problem 48 if the profit on a five-speed bicycle increases from $70 to $110 and all other data remain the same. If the slack associated with any problem constraint is nonzero, find it.In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Animal nutrition. The natural diet of a certain animal consists of three foods: A,B, and C. The number of units of calcium, iron, and protein in 1 gram of each food and the average daily intake are given in the table. A scientist wants to investigate the effect of increasing the protein in the animal’s diet while not allowing the units of calcium and iron to exceed its average daily intakes. How many grams of each food should be used to maximize the amount of protein in the diet? What is the maximum amount of protein? UnitsperGramFoodAFoodBFoodCAverageDailyIntakeunitsCalcium13230Iron21224Protein34560In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Animal nutrition. Repeat Problem 53 if the scientist wants to maximize the daily calcium intake while not allowing the intake of iron or protein to exceed the average daily intake.In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Opinion survey. A political scientist received a grant to fund a research project on voting trends. The budget includes $3,200 for conducting door-to-door interviews on the day before an election. Undergraduate students, graduate students, and faculty members will be hired to conduct the interviews. Each undergraduate student will conduct 18 interviews for $100. Each graduate student will conduct 25 interviews for $150. Each faculty member will conduct 30 interviews for $200. Due to limited transportation facilities, no more than 20 interviewers can be hired. How many undergraduate students, graduate students, and faculty members should be hired in order to maximize the number of interviews? What is the maximum number of interviews?In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution. Opinion survey. Repeat Problem 55 if one of the requirements of the grant is that at least 50% of the interviewers be undergraduate students.Excluding the nonnegative constraints, the components of a linear programming problem can be divided into three categories: the coefficients of the objective function, the coefficients of the problem constraints, and the constant on the right side of the problem constraints. Write a verbal description of the relationship between the components of the original minimization problem 1 and the dual maximization problem 2.The simplex method can be used to solve any standard maximization problem. Which of the following minimization problems have dual problems that are standard maximization problems? (Do not solve the problems.) AMinimizeC=2x1+3x2subjectto2x15x24x13x26x1,x20BMinimizeC=2x13x2subjectto2x1+5x24x1+3x26x1,x20 What conditions must a minimization problem satisfy so that its dual problem is a standard maximization problem?Form the dual problem: MinimizeC=16x1+9x2+21x3subjecttox1+x2+3x3122x1+x2+x316x1,x2,x30Solve the following minimization problem by maximizing the dual problem (see Matched Problem 1): MinimizeC=16x1+9x2+21x3subjecttox1+x2+3x3122x1+x2+x316x1,x2,x30Solve the following minimization problem by maximizing the dual problem: MinimizeC=2x1+3x2subjecttox12x22x1+x21x1,x20Repeat Example 4 if the shipping charge from plant A to outlet I is increased to $7 and the shipping charge from plant B to outlet II is decreased to $3.In Problems 1-8, find the transpose of each matrix. 5327In Problems 1-8, find the transpose of each matrix. 4116In Problems 1-8, find the transpose of each matrix. 178635942In Problems 1-8, find the transpose of each matrix. 913182327In Problems 1-8, find the transpose of each matrix. 2160152013In Problems 1-8, find the transpose of each matrix. 73136109In Problems 1-8, find the transpose of each matrix. 121027801413In Problems 1-8, find the transpose of each matrix. 11321402456138012731In Problems 9 and 10, (A) Form the dual problem. (B) Write the initial system for the dual problem. (C) Write the initial simplex tableau for the dual problem and label the columns of the tableau. MinimizeC=8x1+9x2subjecttox1+3x242x1+x25x1,x20In Problems 9 and 10, (A) Form the dual problem. (B) Write the initial system for the dual problem. (C) Write the initial simplex tableau for the dual problem and label the columns of the tableau. MinimizeC=12x1+5x2subjectto2x1+x273x1+x29x1,x20In Problems 11 and 12, a minimization problem, the corresponding dual problem, and the final simplex tableau in the solution of the dual problem are given. (A) Find the optimal solution of the dual problem. (B) Find the optimal solution of the minimization problem. MinimizeC=21x1+50x2subjectto2x1+5x2123x1+7x217x1,x20MaximizeP=12y1+17y2subjectto2y1+3y2215y1+7y250y1,y20y1y2x1x2P01520510730300121121In Problems 11 and 12, a minimization problem, the corresponding dual problem, and the final simplex tableau in the solution of the dual problem are given. (A) Find the optimal solution of the dual problem. (B) Find the optimal solution of the minimization problem. MinimizeC=16x1+25x2subjectto3x1+5x2302x1+3x219x1,x20MaximizeP=30y1+19y2subjectto3y1+2y2165y1+3y225y1,y20y1y2x1x2P01530510320200531155In Problems 13-20, (A) Form the dual problem. (B) Find the solution to the original problem by applying the simplex method to the dual problem. MinimizeC=9x1+2x2subjectto4x1+x2133x1+x212x1,x20In Problems 13-20, (A) Form the dual problem. (B) Find the solution to the original problem by applying the simplex method to the dual problem. MinimizeC=x1+4x2subjecttox1+2x25x1+3x26x1,x20In Problems 13-20, (A) Form the dual problem. (B) Find the solution to the original problem by applying the simplex method to the dual problem. MinimizeC=7x1+12x2subjectto2x1+3x215x1+2x28x1,x20In Problems 13-20, (A) Form the dual problem. (B) Find the solution to the original problem by applying the simplex method to the dual problem. MinimizeC=3x1+5x2subjectto2x1+3x27x1+2x24x1,x20In Problems 13-20, (A) Form the dual problem. (B) Find the solution to the original problem by applying the simplex method to the dual problem. MinimizeC=11x1+4x2subjectto2x1+x282x1+3x24x1,x20In Problems 13-20, (A) Form the dual problem. (B) Find the solution to the original problem by applying the simplex method to the dual problem. MinimizeC=40x1+10x2subjectto2x1+x2123x1x23x1,x20In Problems 13-20, (A) Form the dual problem. (B) Find the solution to the original problem by applying the simplex method to the dual problem. MinimizeC=7x1+9x2subjectto3x1+x26x12x24x1,x20In Problems 13-20, (A) Form the dual problem. (B) Find the solution to the original problem by applying the simplex method to the dual problem. MinimizeC=10x1+15x2subjectto4x1+x21212x13x20x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=3x1+9x2subjectto2x1+x28x1+2x28x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=2x1+x2subjecttox1+x28x1+2x24x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=7x1+5x2subjecttox1+x24x12x282x1+x28x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=10x1+4x2subjectto2x1+x26x14x2248x1+5x224x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=10x1+30x2subjectto2x1+x216x1+x212x1+2x214x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=40x1+10x2subjectto3x1+x224x1+x216x1+4x230x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=5x1+7x2subjecttox14x1+x28x1+2x210x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=4x1+5x2subjectto2x1+x212x1+x29x24x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=60x1+25x2subjectto2x1+x24x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=35x1+90x2subjectto2x1+5x210x1,x20Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=10x1+25x2+12x3subjectto2x1+6x2+3x36x1,x2,x30Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=10x1+5x2+15x3subjectto7x1+3x2+6x342x1,x2,x30Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=10x1+7x2+12x3subjecttox1+x2+2x372x1+x2+x34x1,x2,x30Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=14x1+8x2+20x3subjecttox1+x2+3x362x1+x2+x39x1,x2,x30Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=5x1+2x2+2x3subjecttox14x2+x36x1+x22x34x1,x2,x30Solve the linear programming problems in Problem 21-36 by applying the simplex method to the dual problem. MinimizeC=6x1+8x2+3x3subjectto3x12x2+x34x1+x2x32x1,x2,x30A minimization problem has 4 variables and 2 problem constraints. How many variables and problem constraints are in the dual problem?A minimization problem has 3 variables and 5 problem constraints. How many variables and problem constraints are in the dual problem?If you want to solve a minimization problem by applying the geometric method to the dual problem, how many variables and problem constraints must be in the original problem?If you want to solve a minimization problem by applying the geometric method to the original problem, how many variables and problem constraints must be in the original problem?In Problems 41 and 42, (A) Form the dual problem. (B) Is the dual problem a standard maximization problem in standard form? Explain. MinimizeC=4x1x2subjectto5x1+2x274x1+6x210x1,x20In Problems 41 and 42, (A) Form the dual problem. (B) Is the dual problem a standard maximization problem in standard form? Explain. MinimizeC=2x1+9x2subjectto3x1x28x1+4x24x1,x20In Problem 43 and 44, (A) Form an equivalent minimization problem with problem constraints (multiply inequalities by 1 if necessary), (B) Form the dual of the equivalent problem. (C) Is the dual problem a standard maximization problem in standard form? Explain. MinimizeC=3x1+x2+5x3subjectto2x16x2x3105x1+x2+4x315x1,x2,x30In Problem 43 and 44, (A) Form an equivalent minimization problem with problem constraints (multiply inequalities by 1 if necessary), (B) Form the dual of the equivalent problem. (C) Is the dual problem a standard maximization problem in standard form? Explain. MinimizeC=25x1+30x2+50x3subjecttox1+3x28x3126x14x2+5x320x1,x2,x30Solve the linear programming problem in Problems 45-48 by applying the simplex method to the dual problem. MinimizeC=16x1+8x2+4x3subjectto3x1+2x2+2x3164x1+3x2+x3145x1+3x2+x312x1,x2,x30Solve the linear programming problem in Problems 45-48 by applying the simplex method to the dual problem. MinimizeC=6x1+8x2+12x3subjecttox1+3x2+3x36x1+5x2+5x342x1+2x2+3x38x1,x2,x30Solve the linear programming problem in Problems 45-48 by applying the simplex method to the dual problem. MinimizeC=5x1+4x2+5x3+6x4subjecttox1+x212x3+x425x1+x320x2+x415x1,x2,x3,x40Solve the linear programming problem in Problems 45-48 by applying the simplex method to the dual problem. Repeat Problem 47 with C=4x1+7x2+5x3+6x4.In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem. Ice cream. A food processing company produces regular and deluxe ice cream at three plants. Per hour of operation, the Cedarburg plant produces 20 gallons of regular ice cream and 10 gallons of deluxe ice cream. The Grafton plant produces 10 gallons of regular and 20 gallons of deluxe, and the West Bend plant produces 20 gallons of regular and 20 gallons of deluxe. It costs $70 per hour to operate the Cedarburg plant, $75 per hour to operate the Grafton plant, and $90 per hour to operate the West bend plant. The company needs to produce at least 300 gallons of regular ice cream and at least 200 gallons of deluxe ice cream each day. How many hours per day should each plant operate in order to produce the required amounts of ice cream and minimize the cost of production? What is the minimum production cost?In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem. Mining. A mining company operates two mines, each producing three grades of ore. The West Summit mine can produce 2 tons of low-grade ore, 3 tons of medium-grade ore, and 1 ton of high-grade ore in one hour of operation. The North Ridge mine can produce 2 tons of high-grade ore, 1 ton of medium-grade ore, and 2 tons of high-grade ore in one hour of operation. To satisfy existing oredrs, the company needs to produce at least 100 tons of low-grade ore, 60 tons of medium-grade ore, and 80 tons of high-grade ore. The cost of operating each mine varies, depending on conditions while extracting the ore. If it costs $400 per hour to operate the West Summit mine and $600 per hour to operate the North Ridge mine, how many hours should each mine operate to supply the required amounts of ore and minimize the cost of production? What is the minimum production cost?In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem. Ice cream. Repeat Problem 49 if the demand for deluxe ice cream increases from 200 gallons to 300 gallons per day and all other data remain the same.In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem. Mining. Repeat Problem 50 if it costs $300 per hour to operate the West Summit mine and $700 per hour to operate the North Ridge mine and all other data remain the same.In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem. Ice cream. Repeat Problem 49 if the demand for deluxe ice cream increases from 200 gallons to 400 gallons per day and all other data remain the same.In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem. Mining. Repeat Problem 50 if it costs $800 per hour to operate the West Summit mine and $200 per hour to operate the North Ridge mine and all other data remain the same.In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem. Human nutrition. A dietitian arranges a special diet using three foods: L,M, and N. Each ounce of food L contains 20 units of calcium, 10 units of iron, 10 units of vitamin A, and 20 units of cholesterol. Each ounce of food M contains 10 units of calcium, 10 units of iron, 15 units of vitamin A, and 24 units of cholesterol. Each ounce of food N contains 10 units of calcium, 10 units of iron, 10 units of vitamin A, and 18 units of cholesterol. If the minimum daily requirement are 300 units of calcium, 200 units of iron, and 240 units of vitamin A, how many ounces of each food should be used to meet the minimum requirements and simultaneously minimize the cholesterol intake? What is the minimum cholesterol intake?In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem. Plant food. A farmer can buy three types of plant food: mix A, mix B, and mix C. Each cubic yard of mix A contains 20 pounds of phosphoric acis, 10 pounds of nitrogen, and 10 pounds of potash. Each cubic yard of mix B contains 10 pounds of phosphoric acis, 10 pounds of nitrogen, and 15 pounds of potash. Each cubic yard of mix C contains 20 pounds of phosphoric acis, 20 pounds of nitrogen, and 5 pounds of potash. The minimum monthly requirements are 480 pounds of phosphoric acid, 320 pounds of nitrogen, and 225 pounds of pottash. If mix A costs $30 per cubic yard, mix B costs $36 per cubic yard, and mix C costs $39 per cubic yard, how many cubic yards of each mix should the farmer blend to meet the minimum monthly requirements at a minimal cost? What is the minimum cost?In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem. Education: resource allocation. A metropolitan school district has two overcrowded high schools and under-enrolled high schools. To balance the enrollment, the school board decided to bus students from the overcrowded schools to the inderenrolled schools. North Division High School has 300 more students than normal, and South Division High School has 500 more students than normal. Central High School can accommodate 500 additional students. The weekly cost of busing a student from North Division to the Central is $5, from North Division to Washington is $2, from South Division to Central is , and from South Division to Washington is $4. Determine the number of students that should be bused from each overcrowded school to each underenrolled school in order to balance the enrollment and minimize the cost of busing the students. What is the minimum cost?In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem. Education: resource allocation. Repeat Problem 57 if the weekly cost of busing a student from North Division to Washington is $7 and all other data remain the same.Repeat Example 1 for MaximizeP=3x12x2+x3subjecttox12x2+x35x13x2+4x3102x1+4x2+5x3203x1x2x3=15x1,x2,x30Solve the following linear programming problem using the big M method: MaximizeP=x1+4x2+2x3subjecttox2+x34x1x3=6x1x2x31x1,x2,x30Solve the following linear programming problem using the big M method: MaximizeP=3x1+2x2subjecttox1+5x252x1+x212x1,x204MPSuppose that the refinery in Example 5 has 35,000 barrels of component A, which costs $25 a barrel, and 15,000 barrels of component B, which costs $35 a barrel. If all other data remain the same, formulate a linear programming problem to find the maximum profit. Do not attempt to solve the problem (unless you have access to software that can solve linear programming problems).